Space of modular forms of level 216, weight 2, and Character 216.d

• Label: 216.2.d
• Level: $216$
• Weight: $2$
• Character orbit: 216.d
• Rep. character: $\chi_{216}(109,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $3$
• Sturm bound: $72$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	216.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(72\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(216, [\chi])\).

	Total	New	Old
Modular forms	42	16	26
Cusp forms	30	16	14
Eisenstein series	12	0	12

Trace form

16 * q + 2 * q^4 + 2 * q^10 + 18 * q^16 - 26 * q^22 - 16 * q^25 - 14 * q^28 + 8 * q^31 - 20 * q^34 - 30 * q^40 + 4 * q^46 + 24 * q^49 - 8 * q^52 - 16 * q^55 + 28 * q^58 - 10 * q^64 + 66 * q^70 - 8 * q^73 + 80 * q^76 - 72 * q^79 - 52 * q^82 + 70 * q^88 - 48 * q^94

Decomposition of \(S_{2}^{\mathrm{new}}(216, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
216.2.d.a	$216$	$2$	216.d	8.b	$2$	$4$	$4$	$1.725$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-6}) \)		✓			216.2.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta_{2} q^{2}-2 q^{4}+(-\beta_{2}+\beta_1)q^{5}+\cdots\)
216.2.d.b	$216$	$2$	216.d	8.b	$2$	$4$	$4$	$1.725$	\(\Q(i, \sqrt{7})\)	None		✓			216.2.d.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{5}+q^{7}+\cdots\)
216.2.d.c	$216$	$2$	216.d	8.b	$2$	$8$	$8$	$1.725$	8.0.629407744.1	None		✓	✓		216.2.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}+\beta _{4}q^{5}+(-\beta _{2}-\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(216, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(216, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 2}\)